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# If the greatest common factor of positive integers n and m is 15, and

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Math Expert
Joined: 02 Sep 2009
Posts: 59588
If the greatest common factor of positive integers n and m is 15, and  [#permalink]

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02 Dec 2019, 00:35
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50% (02:08) correct 50% (02:05) wrong based on 20 sessions

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If the greatest common factor of positive integers n and m is 15, and the remainder when $$(n+x)^{32}$$ is divided by 15 is 1, which of the following CANNOT be the value of x?

A. 1
B. 4
C. 9
D. 11
E. 14

Are You Up For the Challenge: 700 Level Questions

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Joined: 19 Oct 2018
Posts: 1157
Location: India
Re: If the greatest common factor of positive integers n and m is 15, and  [#permalink]

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02 Dec 2019, 17:12
1
GCD(n,m)=15
hence, we can write n=15k

$$(n+x)^{32}$$

= $$(15k+x)^{32}$$

=15$$k^{32}$$ + 32C1 (15$$k)^{31}$$*x+.......+32C1* 15k*$$(x)^{31}$$+ $$x^{32}$$

All the terms are divisible by 15 except the last one. Hence, the remainder when $$(n+x)^{32}$$ is divided by 15 is same as when $$x^{32}$$ is divided by 15

A. 1= (15*0+1)

hence, $$1^{32}$$ will give 1 as a remainder when divided by 15

B. $$4^2$$=16= (15+1)

$$(4^2)^{16}$$= $$(15+1)^{16}$$ will give remainder 1 when divided by 15

D. 11= (15-4)
$$11^{32}$$= $$(-4)^{32}$$ mod 15
$$11^{32}$$= $$(4)^{32}$$ mod 15
$$11^{32}$$= $$1$$ mod 15

E. 14= (15-1)

$$(15-1)^{32}$$ will give 1 as remainder when divided by 1.

C

Bunuel wrote:
If the greatest common factor of positive integers n and m is 15, and the remainder when $$(n+x)^{32}$$ is divided by 15 is 1, which of the following CANNOT be the value of x?

A. 1
B. 4
C. 9
D. 11
E. 14

Are You Up For the Challenge: 700 Level Questions
Re: If the greatest common factor of positive integers n and m is 15, and   [#permalink] 02 Dec 2019, 17:12
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